By using the variational characterisation of the minimiser, see e.g. [1]} for an exposition in the \(\beta \) -ensemble case,
one reduces the construction of \(\sigma _{\mathrm {eq}}^{(N)}\)  to finding a solution to a singular integral equation on the Sobolev space \(H_s( [ a_N \,; b_N ] )\)  driven by the operator
\(\mathcal {S}_N\big [ \phi \big ](\xi ) \; = \; \limits _{a_N}^{b_N} \big ( \mathfrak {w}^{(+)}\big )^{\prime }\big [ \tau _N (\xi -\eta ) \big ] \cdot \phi (\eta ) \mathrm {d}\eta \; .\) 
